Fixed point theorems in hyperconvex metric spaces

Abstract

The notion of hyperconvex spaces was introduced by Aronszajn and Panitchpakdi [1] in 1956. In 1979, independently Sine [13] and Soardi [16] proved the xed point property for nonexpansive maps on bounded hyperconvex spaces. Since then many interesting works have appeared for hyperconvex spaces. For example, see [2–8,14,15]. It is known that the space C(E) of all continuous real functions on a Stonian space E (extremally disconnected compact Hausdor space) with the usual norm is hyperconvex, and that every hyperconvex real Banach space is a space C(E) for some Stonian space E. Then (Rn; ‖ · ‖∞); l∞ and L∞ are concrete examples of hyperconvex spaces. Until recently, the study of hyperconvex spaces was concentrated to the relationship with nonexpansive maps. However, recently, Khamsi [5] established the Knaster– Kuratowski–Mazurkiewicz theorem (in short, KKM theorem) for hyperconvex spaces and applied it to prove an analogue of Ky Fan’s best approximation theorem extending the Brouwer and the Schauder xed point theorems. This seems to be the rst attempt to prove such results in a hyperconvex space setting. In this paper, we obtain a Ky Fan type matching theorem for open covers, a coincidence theorem, a Fan–Browder type xed point theorem, a Brouwer–Schauder–Rothe type xed point theorem, and other results for hyperconvex spaces. Those are usually